Guest Lecture for MIT 18.5096 Topics in Mathematics with Applications in Finance
Jonathan Larkin
October 2, 2025
Disclaimer
This presentation is for informational purposes only and reflects my personal views and interests. It does not constitute investment advice and is not representative of any current or former employer. The information presented is based on publicly available sources. References to specific firms are for illustrative purposes only and do not imply endorsement.
About Me
Managing Director at Columbia Investment Management Co., LLC, generalist allocator, lead Data Science and Research. Formerly CIO at Quantopian, Global Head of Equities and Millennium Management LLC, and Co-Head of Equity Derivatives Trading at JPMorgan.
Fit a model \(f\) to past data by minimizing prediction error: \[
\min_f \sum_t L\big(y_t, f(X_t)\big)
\]
Use \(f\) to predict future outcomes: \(\hat{y}_k = f(X_k), \quad k > t\)
Inputs\(X_t\): features at time \(t\) (e.g., momentum, valuation, sentiment)
Target\(y_t\): what we want to predict (e.g., next-week return)
Model\(f\): linear regression, random forest, neural net
Goal: \(\hat{y}_k \approx y_k\) on new, unseen data
Tasks: regression (continuous \(y\)) or classification (discrete \(y\))
Condorcet Jury Theorem (1785)
If each juror has probability > 1/2 of being correct, and errors are independent, then as the number of jurors n increases, the probability that the majority decision is correct approaches 1. \[
P(\text{majority correct}) \to 1 \quad \text{as } n \to \infty
\]
e.g., sklearn.ensemble.VotingClassifier combines multiple models, but independence/diversity of errors matters.
Boosting Weak Learners (1988)
Kearns, Michael. Thoughts on Hypothesis Boosting. 1988.
Friedman, Jerome H. Greedy function approximation: A gradient boosting machine. 2001.
Sequentially train many “weak learner” models, each focusing on the errors of the previous ones.
Gradient boosted decision trees are the dominant approach in tabular machine learning still today.
\(F_M\) is the ensemble model. After M rounds: \[
F_M(x) = F_0(x) + \sum_{m=1}^M \gamma\, h_m(x)
\]
Each round fits \(h_m\) to the negative gradient of the loss at \(F_{m-1}\), then updates: \[
F_m(x) = F_{m-1}(x) + \gamma\, h_m(x)
\]
\(\gamma\) is the learning rate; \(h_m\) is a weak learner (e.g., shallow tree).
Model Stacking (1992)
Wolpert, David H. Stacked Generalization. 1992.
Train “meta-model” on the predictions of independent base models.
Works best when base models are diverse and capture different aspects of the data.
e.g., sklearn.ensemble.StackingClassifier
Stacking in a Nutshell
Combine several different models by training a meta-model on their predictions.
Train M independent base models \((f_1, \dots, f_M)\) (e.g., linear model, tree, neural net, etc.).
Using an appropriate cross validation scheme, collect out-of-fold predictions for each training example to avoid leakage.
Train a meta-model, \(g\), on these predictions (optionally with the original features). \[
\hat{y}(x) = g\!\big(f_1(x),\, f_2(x),\, \dots,\, f_M(x)\big)
\]
Ensemble Methods Summary
Ensembles reduce variance, improve robustness, and often yield better performance than individual models.
Voting combines models by majority vote.
Boostingsequentially builds models, each correcting the previous.
Stacking combines diverse models, leveraging their strengths.
Model Averaging is a special case of stacking: the meta-model is a weighted linear sum.
These approaches can be combined (e.g., stacking linear model into boosted tree).
The Dunbar Number (1992)
Dunbar, R. I. M. (1992). Neocortex size as a constraint on group size in primates. Journal of Human Evolution, 22(6), 469–493.
Surowiecki, James. The Wisdom of Crowds: Why the Many Are Smarter Than the Few and How Collective Wisdom Shapes Business, Economies, Societies, and Nations. Doubleday, 2004.
For the crowd to be smarter than experts, we require
Diversity of opinion → reduce blind spots
Independence of members → avoid groupthink
Decentralization → empower local knowledge
Aggregation of information → combine insights effectively
The Common Task Framework (2007-)
Donoho, D. (2017). “50 Years of Data Science.” Journal of Computational and Graphical Statistics, 26(4), 745–766.
Define a clear task (e.g., image recognition).
Provide dataset + ground truth labels + hidden test set.
Crypto as an asset class: community builds crypto strategies (Hyperliquid challenge)
Integration with crypto projects:
Fund that stakes in Numerai’s NMR ecosystem
Bridges decentralized funds together
Conclusion
Numerai: pioneered crypto-incentivized crowdsourced hedge fund
Yiedl: DAO-based, on-chain hedge fund built entirely with DeFi
CrowdCent: blends human fundamental research with ML and crypto strategies
Common threads:
Community-driven intelligence
Machine learning aggregation
Cryptocurrency as incentive + infrastructure
Future hedge funds: open, decentralized, global
Human + Machine Collaboration
Types of Collaboration
Horizontal
Vertical
Horizontal
Human forecasts concatenated with features
Fit model on both
Horizontal Example
Cao, S. S., Jiang, W., Wang, J. L., & Yang, B. (2024). “From Man vs. Machine to Man + Machine: The art and AI of stock analyses. Journal of Financial Economics, 160, 103910. https://doi.org/10.1016/j.jfineco.2024.103910
Human forecasts taken as 948,054 twelve-month price forecasts (IBES, 2001–2018) for 6,190 firms by 11,341 analysts
Features for ML model:Firm/industry fundamentals, textual disclosures (10-K, 10-Q, 8-K), and macroeconomic series (FRED)
Target: 12m forward return
Compare performance:
Human only: analyst forecasts
Machine only: an ensemble of ML algorithms (random forests, boosting, LSTM and) trained on the features noted to predict year-ahead stock prices, excluding any analyst forecast inputs
Human + machine: same ensemble trained on analyst forecasts + features
Horizontal Example, Results
The machine only model beats human analyst forecasts 54% of the time
The human + machine model beats human forecasts 57% of the time
Vertical
Stepwise: human first, machine second (or vice versa)
e.g., human generates ideas, machine filters/ranks/optimizes
Vertical Example
deHaan, E., Lee, C., Liu, M., & Noh, S. (2025). “The Shadow Value of Public Information: Evidence from Mutual Fund Managers” (Stanford University Graduate School of Business Research Paper). Available at SSRN.
Constructs 170 alphas from public data (e.g., market, accounting, analysts, text, macro, ratings).
Uses 13-F filing data for 3,337 active, diversified U.S. equity mutual funds (1990–2020) as “human” portfolios.
Portfolio holdings returns are adjusted to subtract style benchmark (aka “DGTW” adjustment); i.e., human + machine is evaluated within fund style/risk/size/liquidity constraints.
AI analyst (just a random forest model!) that
makes an independent stock return forecast, one quarter ahead
uses that forecast to adjust the human manager’s portfolio
AI Analyst Adjustment Methodology
inputs: w_h[j] # human start-of-quarter weight for stock j g[j] # DGTW group of stock j decile[j] # predicted decile within g[j] (1=worst ... 10=best) ŷ[j] # predicted DGTW-adjusted return NAV # fund net asset value at start of quarterinitialize: w_ai := w_h used := ∅ # prevent duplicate use of replacement names# Keep strong human picksfor j in holdings sorted by descending w_h[j]:if decile[j] ==10: used.add(j)
Attempt upgrades for others
# Attempt upgrades for others (largest positions first)for j in holdings sorted by descending w_h[j]:if decile[j] in1..9: group := g[j] C := { s in group | decile[s] ==10and s ∉ used }if C ≠ ∅:# choose best candidate k := argmax_s∈C ŷ[s] target_value := w_h[j] * NAV max_value :=0.20* market_cap(k) delta_value :=min(target_value, max_value) w_ai[k] += delta_value / NAV w_ai[j] -= delta_value / NAV used.add(k)
Replace remaining bottom-decile names
# Replace remaining bottom-decile names with the group indexfor j in holdings:if decile[j] ==1and w_ai[j] >0: idx := index_for_group(g[j]) w_ai[idx] += w_ai[j] w_ai[j] =0# Normalize / cleanproject w_ai onto the simplex (weights ≥ 0, sum=1)
Vertical Example Results
Payout convention: AI’s incremental gain is paid out quarterly so human and AI start next quarter with equal AUM (conservative for AI in dollars).